AO 7667 and the Meaning of ba-an-gi4

§1.AO
7667§
1.1.Scheil§
1.1.1. V. Scheil published in RA 12 (1915) 161-172 a text under the
title “Le calcul des volumes dans un cas particulier à l’époque
d’Ur.” The text is dated to the 4th year of Amar-Sin and
comes, so Scheil, from Umma. He characterized it as “un relevé de
fournitures (de briques), récapitulées d’après
diverses tablettes séparées.” It consists of a
list of receipts of bricks arranged in twelve sections. Ten sections
give linear measures that define volumes and the corresponding number
of bricks, two sections give only the number of bricks. The grand total
amounts to 61,812 bricks. The subscript connects them with the cult
place Gaeš in the vicinity of Ur.

§ 1.2. Neugebauer and Sachs§
1.2.1. The numbers in the text, especially the relationship between
the measurements of lengths, widths, heights, and numbers of bricks
have attracted scholars in the field of Mesopotamian mathematics. O.
Neugebauer and A. Sachs cited it, under the label AO
7667, in connection
with their discovery that the measure “plot” (sar), the
brick sar, as it is usually called, designates a unit of 720 bricks,
which was divided into “shekels” (gin2) of 12 bricks each
(MCT, pp. 95-96). Neugebauer and Sachs were interested in the nine
preserved sections that matched linear measurements with brick numbers.
They say “this text contains nine entries concerning piles of
bricks, four of which we do not dicuss here because of the occurrence
of a term the exact significance of which we are unable to grasp.” Each
entry consists of two sets of linear measurements and one number of
bricks. Neugebauer and Sachs understood the two sets to represent two
piles. They computed their volumes and determined the brick size. It
was 2000 cubic fingers, measuring 20 × 20 × 5 fingers, which they labeled
as type 4.

§ 1.3.Powell§
1.3.1. M. Powell updated the information on various brick types,
re-labeled them and tabulated them with the coefficients that allow
conversion
from volume to number of bricks and viceversa, and extraction of other
data such as weight and number carried by a day laborer over a set
distance (“Metrological Notes in the Esagila Tablet and Related
Matters,” ZA 72 [1982] 119). Type 4 of Neugebauer and Sachs became
type 8 of Powell. He quoted AO
7667 as an example of the usage of brick
metrology “in the regular business of the governing bureaucracy,” specifically “in
architectural planning” (l.c. 117 with note 47).

§ 1.4.Robson§
1.4.1. The “term the exact significance of which” Neugebauer
and Sachs were “unable to grasp” and which thus forced
them to exclude four entries from their consideration is ba-an-gi4.
It appears as qualification of a number within the list of linear measurements
defining volumes. Three times it follows the width, once it follows
the height. It took exactly half a century until E. Robson made sense
of it and published her discovery in her dissertation (see now the
printed version Mesopotamian Mathematics, 2100-1600 BC, Technical
Constants in Bureauceacy and Education (=OECT 14 [1999] 149). She found that
averaging the number qualified by ba-an-gi4 and the preceding number
and using the result as factor results in the volume that corresponds
to the number of bricks. She expressed the physical reality behind
the averaging in the first attestation (see section VI in the tabulation
below) as follows: “Average of 0;30, the base, and 0;05, the
top, (written ba-an-gi4 because the wall tapers or ‘goes back,’ gi4).” On
page 143, she translated ba-an-gi4 as “top (of a tapered wall).” Robson
thought AO
7667 “summarizes records of completed work.” She
did not take issue with the “piles of brick” of Neugebauer
and Sachs, and assumed the volumes were walls.

§1.5.Friberg§
1.5.1. J. Friberg, in “Bricks and Mud in Metro-Mathematical
Cuneiform Texts” in J. Høyrup and P. Damerow, eds., Changing
Views on Ancient Near Eastern Mathematics (=BBVO 19 [2001]),
discussed AO
7667 on pages 136-140. He had learned of Robson’s
discovery from her 1995 dissertation and adopted it for his computations.
He also
gave models for the physical appearance of the volumes. He understood
them as walls consisting of a lower and an upper part, representing
three different configurations: (1) The lower part is 6 1/3 cubits
wide and 2 cubits high and the upper part is 4 cubits wide and 4 cubits
high. The length differs. This is the configuration of the entries
that Neugebauer and Sachs discussed and understood as pairs of piles
of bricks. (2) The upper part tapers. This is the configuration where
two widths are averaged. (3) The end tapers. This is the configuration
where two heights are averaged. Friberg illustrated his models in figure
9.2. Sections of his models for the first two configurations are given
here in figure 1 and 3, and a side view of his model for the third
configuration is given here in figure 5. Note that the legend “5
1/3” of the height on the left side of
the upper part of b4 in Friberg’s figure 9.2 should read “2 1/2.”

§ 2.0.Additional information on AO
7667§ 2.0.1. Scheil thought that the text constitutes a recapitulation of
several separate texts. In the meantime, receipts and promissory
notes that duplicate the measurements and brick counts of AO
7667 have indeed
been published. A synopsis of the texts that have come to my attention
is given in table 1.

§ 2.1.Brick-piles

UTI 5,
3394, after duplicating
the measurements of section VI a and b, states: ša3 SIG4.ANŠE
1-a-kam / sig4 u3-ku-ru-um-bi 17 1/2 sar “They
(the measurements) are of 1 brick pile. Its baked bricks (amount to)
17 1/2 brick sar.” PDT 2,
1377, after duplicating the measurements of section V a and b, states:
SIG4.ANŠE ar-ha u3-ku-ru-um-ma “Brick
pile of half-bricks and baked bricks.” Generalizing the evidence
of these texts we can conclude that each section of AO
7667, except
VII
and XII, represents the measurements of one brick pile and the count
of its bricks.

§ 2.2.Half-bricks§
2.2.1. Before considering the relevance of the presence of half-bricks
in the brick piles, I make a detour on the word written ar-ha and related
matters. The word appears to be an old, that is pre-Ur III, loan from
Akkadian arhum. Akkadian arhum means also “cow.” The
dictionaries understand arhum “cow” and arhum “half-brick” as
homonyms, but the OB logographic writing SIG4.AB2 for arhum “half-brick” is
sometimes seen as indicating a single word “cow,” which
would designate, for unknown reasons, also the half-brick. W. von Soden
writes in AHw arhu III “nach Wortzeichen = arhu II, also Kuhziegel?” Robson, 58, states “‘half-bricks,’
literally ‘cows,’” and
Friberg translates “cow-bricks.” The word arhum “cow” is
feminine, but masculine gender is attested for arhum “half-brick” if
sig4ar-hu in the Old Akkadian text ITT 5,
9322 spells the plural
arhū. If so, arhum “cow” and arhum “half-brick” are
homonyms. Sumerian may not have had its own word for half-brick
and borrowed it from Akkadian early on, and scribes invented the
logographic
spelling SIG4.AB2 on the basis of this homonym.

§
2.2.2. M. Sauvage (personal communication) makes the point that “half-bricks
appeared with square bricks (half-bricks are always used with square
bricks) in central and southern Mesopotamia with the Akkadian period
only. During the whole Early Dynastic period in central and southern
Mesopotamia, one used only rectangular bricks (flat or more often
plano-convex) and no square bricks. The square bricks (and half-bricks)
seem to appear
and to become widespread in northern Mesopotamia at the latest
during the ED III period and to reach central and southern Mesopotamia
during
the Akkadian period. Thus, one can consider that Sumerians needed
no word for half-bricks until the Akkadian period. They could have
borrowed
the word from the Akkadian populations of Upper Mesopotamia when
the use of square bricks and of half-bricks became widespread in
the South.”

§
2.2.3. The same thing appears to have happened to the word for brick-pile,
which is amārum (not amarum, as posited in the dictionaries,
because of Proto Izi I 261-262 and 267 [MSL XIII 26]: SIG4.ANŠE
a-ma-a-rum, SIG4.DU3a-ma-a-rum) in Akkadian, and was written pseudo-logographically
SIG4.ANŠE on the basis of the near-homonymity imāru “donkey” and
amāru “brick-pile,” as suggested already
by Robson, 67. I would expect that Sumerian borrowed the Akkadian
word, probably
in the form *amara.

§ 2.2.4. Returning to the designation of the bricks of
section V as “half-bricks” (ar-ha)
and “baked (square) bricks” (u3-ku-ru-um),
it appears likely that all brick-piles of AO
7667 consisted of
an admixture
of half-bricks.
They were needed in any case for corners, ends and other special
areas of the future brickwork. In the count, two were counted as
1 full brick
(u3-ku-ru-um).

§ 2.3.Models of the brick piles§
2.3.1. Friberg’s models of what he believed to be brick walls
are not the only solutions. Since the measurements describe brick piles,
it appears more natural to center the upper parts above the lower parts
(see figures 2, 4, and 6). I have sketched models in figures 7-9 of
how the upper part of the brick piles might have been stacked. Figure
7 shows a section of a brick pile of the first configuration according
to Friberg’s model, figure 8 according to my alternative.
Figure 9 shows the central column and the left half of the upper
part of
the third configuration. In constructing the models, I have used
only half-bricks
were they are needed to conform to the measurements of the upper
part of tapering brick piles. I suspect that many more half-bricks
were
included. I assume that the bricks were laid down rather than
stood on end, and, following a suggestion of R. Englund, have
avoided
the formation of columns for added stability.

§ 2.4.The brick count§
2.4.1. Looking at the models of the sections and the side view of brick
piles, an explanation or the absence of any record of the volumes in
AO
7667 appears. Friberg, wondering about that lack, said “since
the volumes are not explicitly mentioned in the text, they must have
been recorded somewhere else.” Neugebauer and Sachs,
Robson, and Friberg assumed that the ancient scribe, like them,
measured
the bodies of brick, computed their volumes and converted the
volumes into
numbers of bricks by applying the appropriate coefficient for
the type-8 brick (2;42). When comparing the calculated volumes
with
the numbers
of bricks given in the text, they noted discrepancies in 5
cases. I give in table 2 below the number of bricks, and add
the sexagesimal
notation in parentheses for easy comparison with Robson and Friberg.

section

# of bricks corresponding to calculated volume

# of bricks in text

missing bricks

III

6,966 (9;40,30)

6,960 (9;40)

6

VI

12,636 (17;33)

12,600 (17;30)

36

VIII

2,322 (3;13,30)

2,316 (3;13)

6

IX

20,586 (28;35,30)

20,520 (28;30)

66

X

4,116 (5;43,07,30)

4,104 (5;42)

13

Table 2

§2.4.2. Neugebauer and Sachs said of the discrepancy in sections
III and VIII “apparently in order to avoid fractions of the gin2.” Robson
called the number of bricks in the text “rounded figures” and
contrasted them with the “correct value.” Friberg also
thought that the scribe “rounded to the nearest shekel” in
sections III and VIII. He suspected a mistake of Scheil’s copy
in section VIII and did not comment on the discrepancies of sections
IX and × .

§ 2.4.3. I believe the volumes do not appear in the text because the
scribe never computed them. He counted the bricks that were visible
at one end of the pile. What he saw represented 20 fingers, or 2/3
of a cubit, of the length of the pile. He measured the length of the
pile, computed the number of 2/3-cubit sections and multiplied it with
the bricks visible. If there were bricks missing from the pile, which
was easy to see, he subtracted them.

§ 2.5.Administrative aspects of AO
7667§ 2.5.1. Neugebauer and Sachs, Robson, and Friberg were interested in
the metrological aspects of AO
7667. The text is also interesting
in other aspects, especially as source for the execution of a royal construction
project in the Ur III kingdom.

§ 2.5.1.1.The persons of the formula DUB PN
Each of the 12 sections of the text is subscribed with the formula
DUB PN, which means literally “tablet of PN” when DUB is
read dub, or “seal of PN” when DUB is read kišib3.
In either case, the subscript designates a tablet on which PN rolled
his seal and therewith recorded the receipt of the goods mentioned
in the text of the tablet. The recipient and the receiver are in most
cases the same person. But the document PDT 2,
1377, which duplicates
section V, states that the bricks were received (šu ba-ti) by
Lukala, the executive officer (šabra) of “dNin-uru-a-mu-DU,” and
that (the receipt) was sealed with the seal of Girini-isa (kišib3
giri3-ni-i3-sa6-ga ib2-ra), while Girini-isa appears in the formula
DUB PN and Lukala goes unmentioned in the account AO
7667.

§ 2.5.1.2. AO
7667 was not sealed, but the additional
sources were sealed. The seal legends show that Girini-isa, Dayyan-ili,
Lukala, and Šulgi-ezen
were scribes. Lukala was in addition an executive officer (šabra).
Iddin-Ea (i-ti-e2-a) was according to his legend a “servant
of Enki.” The other persons appearing in the subscript DUB
PN may or may not have been scribes. Šu-lulu was employed by
the god Ningublaga, the city of Sippar, and the governor of Adab.

person in formula DUB PN

employer

section

Ku-Ningal

-

I

ARAD-Nanna

executive officer of dNin-[ ]

II

Beliya

executive officer of An

III

Iddin-Ea

Enki

IV

Girini-isa

dNin-a-mu-DU

V

Lukala

dNin-a-mu-DU

VI

Šulgi-ezen

dNin-a-mu-DU

VII

Šu-lulu

Ningublaga

VIII

Šu-lulu

Sippar

IX

Dayyan-ili

governor of Adab/Adab

X/PDT 2, 1353

Šu-lulu

governor of Marad

XI

Riš-beli

governor of Kazallu

XII

Table 3

§2.5.2.The employers of the persons in the formula DUB PN§
2.5.2.1. They are mentioned in the subscript of each section of the
text after the name of the person in the formula DUB PN. The relationship
between employee and employer is designated as “man of” (lu2).
The employee in section II is identified as “brother of the executive
officer of dNin-[ ], which may or may not mean that the executive officer
of dNin-[ ] was the employer.

§ 2.5.2.2. The table is split into two divisions. A certain Lugal-magure
was the responsible official for sections I-VIII, Lu-dingira for sections
IX-XII.

§ 2.5.2.3. The gods Enki and Ningublaga were the immediate
employers of the persons in the formula DUB PN. The god An employed
Beliya through
his executive officer. dNin-a-mu-DU was probably a person. A god of
that name is not attested. There exists a personal name dŠara2-a-mu-DU.
A. Limet, L’Anthroponymie Sumerienne dans les documents de la
3e dynastie d’Ur (Paris 1968) 309, understands it as Šara2
+ agentive + mu-tum2 and translates “Šara a apporté,” but
the -a in dNin-a-mu-DU shows that his interpretation cannot be correct.
A complication is introduced by PDT 2,
1377, where the name is transliterated
as dNin-uru-a-mu-DU and by SNAT346 where it is transliterated as dNin-ha!-mu-DU.
dNin-a-mu-DU was perhaps a representative of one or more gods.

§ 2.5.2.4. The employers in the lower division, probably including section
IX, were governors.

§ 2.6.A model for the function of AO
7667§
2.6.1. The brick piles listed in AO
7667 were received by gods of cities
of Sumer and a governor of a city of Sumer, Adab, and governors of
cities of Akkad. So the whole kingdom seems to have been involved.
This indicates organization on the royal level, which is confirmed
by the use of the “imperial” calender in PDT 2,
1353,
1370
and
1377. SNAT346 has the subscript “Gaeš Gipar house” (e2
gi6-par3 ga-eški), PDT 2
1353 “to the Gaeš Gipar
house” (e2 gi6-par3 ga-eški-še3). In AO
7667 rev.
ii 16, ša3 ga-eški, is visible at the end of the line.
It may have been indented, in which case the subscript was just “in/of
Gaeš.” Otherwise, [e2 gi6-par3] can be restored. The bricks
were apparently destined for the construction of the Gipar house in
Gaeš on the quay Karzida that is celebrated in the inscriptions “Amarsuen
6, 8, and 11” (see H. Steible, FAOS 9/2 [Stuttgart 1991] 229-238).

§ 2.6.2. One expects that the bricks came from various
parts of the kingdom and were used to build the residence, yet the
bricks were received
by representatives of clerical and secular powers throughout the kingdom.
The bricks could hardly have come from all over the kingdom to be distributed
throughout the kingdom. The key to resolving this problem may be found
in the role that the governor of Umma played in the project. SNAT346
states that Šulgi-ezen was obliged to ‘return’ the
bricks that had come from the governor of Umma. UTI 5,
3394 states
that Lukala swore to ‘return’ the brick-pile of 17 1/2
brick sar, specifying that these were “bricks of the governor
of Umma.” He was obliged to ‘return’ the bricks on
the first day of the month following the month when the brick piles
of sections IV and V, and possibly those of all other sections, including
VI, had been received.

§ 2.6.3. It is inconceivable that large numbers of bricks
that were received in one month had to be returned in the next. The
problem surely lies
with the translation of gi4 “to return.” In PDT 2,
1353,
we find ki-ba gi4 instead of simple gi4. ki-ba gi4 is close to ki-be2
gi4, which means “to restore,” and ki-ba gar, which means “to
replace.” I suggest that gi4 and ki-ba gi4 in the present context
mean in effect “to reimburse.”

§ 2.6.4. These considerations lead me to the following
scenario: the king instructed the magnates of his land to deliver the
bricks necessary
for building the residence of the En priestess of Nanna in Gaeš.
They had the option of letting the governor of Umma deliver their share
of bricks, as long as they promised to reimburse him, presumably with
goods and services that they could better afford. The deadline for
the delivery of bricks to the construction site was the month Akiti
of AS 4. The representatives of those who had elected to have their
share of bricks delivered by the governor of Umma, went to the construction
sites themselves, or sent their scribes, counted their share of bricks,
took possession of them, and handed them over to the royal administration.
The representative of the governor of Umma received from them notes
promising immediate reimbursement. The individual receipts of the bricks
that had been delivered by the governor of Umma were collected and
copied on the tablet AO
7667, presumably to serve as record for the
governor of Umma.

§ 3.Additional references for the term ba-an-gi4§ 3.1.NES 48-06-103§
3.1.1. The text belongs to the collection of the Department of Near
Eastern Studies, Cornell University. It will be published soon by D.
I. Owen and R. Mayr together with other administrative tablets that
record activities in the little known town of Garšana. I thank
D. I. Owen, Curator of Tablet Collections, for allowing me to quote
and for providing me with access to this unique archive. The occurrence
of the term ba-an-gi4 in this text lead to the present article.

section

length

width

bangi

height

volume in sar.gin2.še

I a

96

4

2

5.20

I b

96

3

2

4

6.40

II

90

3

2

5

7.51.45

III

82

2 1/2

2

4

5.07.90

ugula Ba-zi

IV

21

4

2

1.10

V

66

3

2

5

5.43.135

ugula Simat-E2-a

Table 4

§3.1.2. The top of NES 48-06-103 is broken, so beginning and
end are missing. The preserved part of the text gives the linear measures
and volumes of mudwalls (im-du8-a). The subscripts identify the overseers
whose workers built the mudwalls. Only the fully preserved sections
are documented in table 4. I give all linear measures in cubits.

§ 3.1.3. The bangi are listed after the width, which means
that the walls tapered toward the top. Robson’s formula, length × ((width+bangi)÷2) × height, yields the volumes given in the text. In three cases the
volume resulting from the measurements is calculated down to volume
grains (še), that is 1/180 of a shekel, and not rounded to the
shekel or otherwise. The tapering top of the mud walls may have had
the form Friberg assumes for the brick-piles (figure 3) or the alternative
suggested by me (figure 4).

§ 3.1.4. Instead of ba-an-gi4, the scribe wrote ba-an-gi4-bi. In section
I b, he had written ba-an- when he reached the end of the line. He
erased ba-an-, probably intending to indent it. But he forgot, and
indented only gi4-bi.

§ 3.2.YOS 1,
24§
3.2.1. The text is the record of a survey of 5 garden plots. A plan
is drawn on one side of the tablet. It shows an essentially rectanglar
frame. The lengths of the sides and the identification of the adjacent
land are written outside and along its four sides. Three sides are
16 poles long, the fourth side 14 poles and 4 cubits. Within the frame
is an empty area that is surrounded by triangles and quadrilaterals.
The lengths of their sides and the extent of their surfaces is written
inside and along their sides. The triangles and quadrilaterals are
called “terrains” (ki). The empty area that is surrounded
by theses terrains represents the 5 garden-plots. The surveyor did
not bother to sketch them individually. The other side of the tablet
is inscribed with the text. This side is called the reverse by the
editor, A. T. Clay, which may or may not be born out by the shape of
the tablet, but its two columns are arranged in the manner of an obverse.
The text has the form of a balanced account. The “capital” is
presented as the measurements of the frame and the surface enclosed
by it. The frame is described as follows: 16 ninda us2 16 ninda sag
14 ninda 4 kuš3 ba-gi4 “16 poles long side, 16 poles front,
14 poles 4 cubits bangi.” The enclosed surface is given as 49
2/3 surface sar 1 shekel “terrain” (ki) and 193 surface
sar “field” (a-ša3). From it (ša3-bi-ta),
the field and the terrains are deducted. The “field” corresponds
to the 5 garden plots that are detailed in the second section. 4 measure
38 sar 36 shekels, one 38 sar 34, or 35, or perhaps even 36 shekels.
Clay, who is an exceedingly trustworthy copyist, shows a clear 4 and
the stunted head of a vertical wedge set a little higher than the upper
vertical wedges that write the numeral 4. The next sign, which is gin2,
appears somewhat cramped. The copy gives the impression that the scribe
corrected the 4 to a 5 or a 6. 5 × 38 sar 36 shekels = 193 sar, the
one number of the “capital.” The terrain of the capital
is the total of 6 terrains inscribed on the plan. It is impossible
to verify the sum of their surfaces from the plan. The numbers of
the upper 3 terrains are fully preserved, namely 10 1/2 sar, 3 1/2
sar,
and 12 sar; the number of the terrain on the right side is partly
preserved and appears to be 16 2/3 sar 6 1/3 shekels. The surface
of these 4
terrains is then 42 sar 46 1/3 shekels. The feeble traces the numbers
of the 2 terrains along the lower sides are unreadable. They should
have measured 6 sar 34 2/3 shekels.

§ 3.2.2. The 193 sar of garden plots and the 49 sar 41
shekels terrains amount to a total surface of 242 sar and 41 shekels.
How did the surveyor
harmonize this surface with the measurements of the frame that encloses
garden plots and terrains? He appears to have squared the long sides,
that is 16 poles × 16 poles = 256 sar. Then he subtracted the short
side from a long side, 16 poles - 14 poles 4 cubits = 1 pole 8 cubits,
muliplied this with a long side, which yields 26 sar 40 shekels,
and divided this by 2, thus 13 sar 20 shekels. This surface he subtracted
from 256 sar and arrived at 242 sar 40 shekels. His computation is
based on the wrong premise that the triangle which he subtracted
was
rectangular. The three equal sides of the frame imply the form of
a trapezoid (see figure 10). In order to accurately calculate its surface,
the scribe would have had to determine the length of the longer of
the short sides of the triangles that coincided with the longer side
of the core-rectangle of the trapezoid. To do that, he would have
had
to use the Babylonian equivalent of the Pythagorean principle (see
P. Damerow, “Kannten die Babylonier den Satz des Pythagoras,” BBVO 19 [2001] 232-238). If he knew it, the calculations might have been
too tedious for him. It is more likely that he did not see the frame
as a trapezoid, but rather as a square with a triangular section
missing (see figure 11).

§ 3.2.3. The terrain in the upper right corner of the plan also has 3
equal sides of 3 1/2 poles and a fourth side of 3 poles, and must
have been a trapezoid, too. In this case, the surveyor used a different
method of calculation. He multiplied the short side with a long side,
that is 3 1/2 poles × 3 poles = 10 1/2 sar, then he deducted the
short side from the longside and multiplied the result with the short side,
that is 1/2 pole × 3 poles = 1 1/2 sar, and combined the two results
to arrive at 12 sar.

§ 3.2.4. Returning to the total surface of his plan, the calculation
of the surface from the lengths of the four sides of the plan by
the surveyor is one shekel short of the total of garden plots and terrains.
Presumably, he added up the individual surfaces of terrains and gardenplots
and came up with 242 plots and 41 shekels. But when he computed the
entire surface from the measurements of the frame, he came up with
242 sar and 40 shekels.

§ 4.The meaning of ba-an-gi4§
4.1. Based on the references of ba-an-gi4 in AO
7667, Robson suggested
as meaning of the word “top (of a tapered wall).” In view
of the 3rd configuration (see §1.5 above), Friberg expanded this
meaning to “top or end of a tapered wall.” The appearance
of the term in the two-dimensional context of YOS 1,
24, shows
that the actual meaning must still be more general.

§ 4.2. If we paraphrase the physical form that is associated
with the term, it designates the tapering from the value mentioned
immediately
before ba-an-gi4 to the the value that is qualified by ba-an-gi4.
6 kuš3 dagal 1 kuš3 ba-an-gi4 designates a tapering from
a width of 6 cubits to a width of 1 cubit, 2 1/2 kuš3 sukud
1 kuš3 ba-an-gi4 a tapering from a height of 2 1/2 cubits to
a height of 1 cubit, and 16 ninda sag 14 ninda 4 kuš3 ba-gi4
a tapering from a length of 16 poles to a length of 14 poles 4 cubits.
If we paraphrase the method of calculation accociated with ba-an-gi4,
ba-an-gi4 designates the averaging of the two values and using the
result as factor in a multiplication, and if we paraphrase the geometrical
conception that corresponds to this method of calculation–and
so links it with with the physical form–, ba-an-gi4 designates
the rectangle or rectangles that are double the surfaces of the
rectangular triangles whose hypotenuse constitutes the angle of
tapering. It
may be expected that the actual meaning of ba-an-gi4 is descriptive
of
one of these three paraphrases. Robson and Friberg proposed that
it was descriptive of the physical form.

§ 4.3. The etymological meaning of the word ba-an-gi4
is a problem. The standard translation of the verbal base gi4 is “to
return,” which
is based on the common equation with Akkadian târu.
Robson translated “goes
back,” Friberg, with questionmark, “returned.” But
nothing in the physical shape appears to return or go back. In Nabnitu 12,
172 (MSL 16, 196), gi4 is equated with ekēmu.
When used transitively, the Akkadian verb means usually “to
take away,” but when
describing parts of the liver, especially protruding forms as the
processus caudatus (“finger”) and processus papillaris
(maš2),
it describes some configuration. R. Leiderer, Anatomie der Schafsleber
om babylonischen Leberorakel, (Munich 1990) gives many examples.
CAD translates “to be stunted, atrophied.” If this
meaning is posited for gi4 in ba-an-gi4,
it is a small the step to “taper,
bevel, slant, incline.” Of course, while being formally a
finite verbal form, ba-an-gi4-bi “its ba-an-gi4” in
NES 48-06-103 shows that it was a noun. So we might translate it
as the noun “taper,
bevel, slant, incline.”

§ 5. In the following, a tabulation of AO
7667 with interspersed
comments is offered. The column “section” gives Scheil’s numbering
first, Robson’s second, Friberg’s third, and mine fourth.
Linear measures are given in cubits, bricks in sar and gin2.

DUB šu-lu-lu lu2 sipparki
Scheil’s copy shows ud-x-(x) ki, where x (x) looks like lugal.
I speculate that x = KIB and (x), which consists of an oblique wedge
with 3 superimposed winkelhaken, is the sign NUN that was written obliquely
because wrapping the sign around the edge horizontally would have run
into space on the obverse that was already occupied by writing. The
volume corresponds to 28.35 1/2 and a fraction, so 66 bricks were missing
from the brick-pile.